The coordinates \((8, 10)\) and \((-2, 4)\) represent two points in a Cartesian coordinate system. There are several mathematical calculations we can perform with these points, such as finding the distance between them, the midpoint, or the slope of the line connecting them.

Let's go through each of these calculations:

1. **Distance between the two points**:
   The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]

   Substituting the given points \((8, 10)\) and \((-2, 4)\):
   \[
   d = \sqrt{((-2) - 8)^2 + (4 - 10)^2}
   \]
   \[
   d = \sqrt{(-10)^2 + (-6)^2}
   \]
   \[
   d = \sqrt{100 + 36}
   \]
   \[
   d = \sqrt{136}
   \]
   \[
   d = 2\sqrt{34} \approx 11.66
   \]

2. **Midpoint between the two points**:
   The midpoint \(M\) of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the midpoint formula:
   \[
   M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
   \]

   Substituting the given points \((8, 10)\) and \((-2, 4)\):
   \[
   M = \left(\frac{8 + (-2)}{2}, \frac{10 + 4}{2}\right)
   \]
   \[
   M = \left(\frac{6}{2}, \frac{14}{2}\right)
   \]
   \[
   M = (3, 7)
   \]

3. **Slope of the line through the two points**:
   The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the slope formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

   Substituting the given points \((8, 10)\) and \((-2, 4)\):
   \[
   m = \frac{4 - 10}{-2 - 8}
   \]
   \[
   m = \frac{-6}{-10}
   \]
   \[
   m = \frac{3}{5}
   \]

### Summary:
- **Distance**: \(2\sqrt{34} \approx 11.66\)
- **Midpoint**: \((3, 7)\)
- **Slope**: \(\frac{3}{5}\)

Would you like more details on these calculations or have any questions?

Here are some related questions you might find interesting:
1. How would you find the equation of the line that passes through these two points?
2. What is the equation of a circle with these two points as the endpoints of its diameter?
3. How do you determine if three points are collinear?
4. What are the properties of the line segment that joins these two points?
5. How can you calculate the area of a triangle formed by these two points and the origin?

**Tip:** The slope of a line is a measure of its steepness, and it can tell you how much the line rises or falls as you move from left to right.

Learn how to calculate the distance, midpoint, and slope between the points (8, 10) and (-2, 4) in a Cartesian coordinate system. Understand the applications of the distance formula, midpoint formula, and slope formula with detailed step-by-step solutions. Suitable for high school students.

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